The transverse and longitudinal current correlations in a simple classical fluid were originally calculated from a modeled kinetic equation using an approximate solution valid for only small values of wave number $k$. These correlation functions have been recalculated using the exact solution of this kinetic equation for arbitrary values of $k$.

A modification of a previously discussed model for condensation in Fermi systems is applied phenomenologically to ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The modified model contains an infinite sum of simple many-body interactions. Upon comparison of the model with experiment, good agreement is obtained for the exponents ${\gamma }^{\prime }$, $\beta$, and $\delta$. The exponent ${\alpha }^{\prime }$ describes a slightly stronger singularity than is observed, and the exponents $\gamma$ and $\alpha$ retain their classical values.

Eigenphases near a resonance are shown to follow an equation which is a simple generalization of the usual equation for the behavior of the phase shift near a resonance in single-channel scattering. The parameters in the equation are shown to have a simple graphical interpretation.

This note presents data on liquid metal-ammonia solutions that were omitted from an earlier tabulation and discussion of the transport characteristics of electronically conducting liquids. Similarities and differences between the metal-ammonia solutions and the liquids included in the earlier publication are noted and briefly discussed.

The response of a single-spin system in the NMR condition to irradiation with two strong electromagnetic resonant waves has been observed. Multiple quantum lines involving up to 27 quanta have been detected. The interpretation of the spectrum by the second-quantization formalism is given.

Numerical spin-unrestricted Hartree-Fock (SUHF) wave functions have been computed for the neutral open-shell atoms Li to Br and the transition-metal ions ${\mathrm{Sc}}^{2+}$ to ${\mathrm{Cu}}^{2+}$. These wave functions have been used to obtain accurate values of the SUHF spin densities. These calculations are used to make a critical review of the numerical accuracy of previously reported SUHF spin densities and to compare our accurate values of the SUHF spin densities with those extracted from experimental data. A large majority of previous SUHF calculations have been performed using the analytic-expansion method. The spin density at the origin computed from analytic-expansion calculations is sensitive to the choice of basis functions, and it is very difficult to obtain a precise measure of its accuracy. Our numerical SUHF calculations yield the SUHF value of the spin density $\chi$ to an accuracy conservatively estimated to be ±0.01 a.u. Comparison with analytic-expansion calculations indicates that the use of large, carefully selected basis sets usually yields spin densities within 20% of the numerical SUHF values reported here. The calculated SUHF spin densities are compared with available experimental values and it is found that (a) for the alkali atoms Li, Na, and K, the SUHF spin densities are 97, 86, and 77% of experiment; (b) for the first-row atoms N,O, and F, the SUHF spin densities are 193, 172, and 186% of experiment; (c) the SUHF spin density for P is opposite in sign and 153% of the absolute value of the experimental spin density; (d) the SUHF spin densities of the $4s^{2}3d^n$ transition-metal atoms Ti, V, Mn, and Co are 82, 75, 72, and 67% of experiment; and (e) the SUHF spin density of As is 231% of experiment. It is concluded that, although some correlations are possible between SUHF and experimental values, the SUHF method should be used with extreme caution for the prediction of spin densities at the origin. This work provides a guide to the reliability of the SUHF predictions.

The solution to the nonrelativistic Schrödinger equation for a one-electron ion in a generalized central-field potential is investigated using high-order perturbation theory. It is shown that by utilizing a finite expansion of the perturbation-theory wave function in terms of associated Laguerre polynomials, perturbation-theory results can be obtained for any $n$, $l$ state to arbitrarily high order. Results for the wave function and energy are explicitly given to third and fourth order, respectively. It is also shown that by reexpressing the high-order perturbation-theory energy expansion as a series of rational fractions (Padé approximants), accurate eigenvalues are obtained, even for large values of the expansion parameter.

A general and formally exact class of integral-transform electronic wave functions may be defined in terms of a distortion operator. This operator, which depends on one or more running parameters, maps some zero-order approximate wave function onto any part of a Hilbert space. The set of all such distorted functions may then completely span the space. The most effective discrete set of $n$ configurations for the approximation of the exact integral transform is considered from the standpoint of a numerical analysis. It is taken to be the set which permits the most accurate $n$-point evaluation of an integral expression for the expectation value of the energy. In practice, the diophantine integration scheme may be used to provide such a set of configurations. The value of the optimal transform weighting function for ground and excited states at each of the $n$ points can be inferred back at the end via the solution to a conventional linear matrix eigenvalue equation. Preliminary applications are made to the calculation of the radial correlation limits for the helium, lithium, and beryllium atoms.

A calculation is made of the radiative electric dipole transition probabilities coupling the vibrational levels of the $2p\pi {}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ and the $2s\sigma {}_{}{}^3{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}$ electronic states of hydrogen. The radiative lifetimes for those vibrational levels lying above the ground state are found to be approximately 100 μsec. The lifetime of the ground vibrational level, decaying by magnetic dipole and electric quadrupole emissions, is estimated to be 1 msec. Finally, a calculation is made of the dependence of the radiative lifetimes upon an external electric field.

The $K$-shell fluorescence yield ${\omega }_K$ has been measured for the low-atomic-number elements beryllium, boron, carbon, fluorine, and magnesium. The primary vacancies in the $K$ shell were produced by an intense beam of $K$ x rays generated by electron bombardment of aluminum and carbon targets. The measured values of ${\omega }_K$ agree quite well with values calculated from a theoretical prediction of Wenzel. They exhibit only fair agreement with semiempirical formulas which include screening and relativistic effects, and with a recent calculation by McGuire based on the $K$-shell Auger transition rate.

The microwave double-photon transitions $J=3\mathrm{}\twoheadleftarrow 2$ and $J=2\mathrm{}\twoheadleftarrow 0$ have been observed for the symmetric-top molecules C${\mathrm{D}}_3$CN and P${\mathrm{F}}_3$ using high-power microwave double resonance. The pumping of the $J=2\mathrm{}\twoheadleftarrow 0$ transition of C${\mathrm{D}}_3$CN was so efficient in the low-pressure region that a negative temperature was established between the $J=1\mathrm{}$ and the $J=0\mathrm{}$ levels, and induced emission $J=1\mathrm{}\to 0$ was observed. The pressure variation of the efficiency of the double-photon pumping agreed quantitiatively with theoretical values calculated using the Göppert-Mayer formula for the relevant dipole matrix element and Karplus and Schwinger's formula for saturation. The experiment provides interesting information on collision-induced transitions in C${\mathrm{D}}_3$CN.

Pure correlation effects on the hyperfine structure of atomic boron were investigated and found to be important. A natural spin-orbital expansion of the well-correlated 187-term wave function of Schaefer and Harris was used. Pure correlation effects are those contributions to an expectation value obtained when the wave function is improved beyond the best possible Slater determinant (the best overlap determinant). Our results indicate that the unrestricted Hartree-Fock method yields valuable information about the orbital and spin magnetic dipolar and electric quadrupolar terms, but gives unreliable results for the Fermi contact term. Utilizing both experimental and theoretical results, the Fermi contact term $f$ was estimated as $0.096{a_0}^{-3\mathrm{}}$, and the quadrupolar nuclear-shielding factor $\gamma$ as 0.093.

The spontaneous emission of a set of $N$ identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.

Atomic form factors and incoherent-scattering functions for the lithium and beryllium isoelectronic sequences through $Z=8\mathrm{}$ have been calculated with correlated ground-state wave functions. The form factors differ from Hartree-Fock values by 1-2% and differ from Hartree-Slater values by as much as 10%. The incoherent-scattering functions are much more dependent on correlation, with the values for neutral beryllium differing from the Hartree-Fock values by as much as 30% at very small values of the momentum transfer.

A study of several statistical approximations to the Hartree-Fock (HF) one-electron effective exchange potential has been carried out through a series of restricted and spin-polarized Hartree-Fock-Slater (HFS) calculations on $3d$-transition-row atoms and ions. The results of this study for the ${\mathrm{Mn}}^{+2\mathrm{}}$ ion are reported in this paper and are compared with the corresponding HF results. Of the various approximations in which only the homogeneous part of the local exchange potential is kept, the simple $\mathrm{X}\alpha$ method is shown to give quite good results. This method amounts to multiplying the exchange potential $-6\mathrm{}{\left[\right(\frac{3}{4\pi }\left){\rho }_{\xi }\mathrm{}\right(r\left)\right]}^{\frac{1}{3}}$, where $\xi =\pm \frac{1}{2}$, by a constant $\alpha$ which is usually determined by the requirement that the $\mathrm{X}\alpha$ orbitals correspond to a minimum in the HF energy. It is shown that the energy-dependent exchange schemes proposed earlier break down totally in the spin-polarized case. We examine the importance of including the inhomogeneous correction terms via the method recently proposed by Herman et al. and we discuss the possible importance of these terms in spin-polarized energy-band calculations. We conclude that the $\mathrm{X}\alpha$ method is both accurate and simple enough for practical use in carrying out energy-band calculations for magnetic solids.

Angular correlations between $K$ and $L$ x rays in Pb were measured. NaI(T1) and Ge(Li) detectors were used for the detection of $K$ and $L$ x rays, respectively. The angular correlation functions for the $K-L\alpha$, $K-L\beta$, and $K-\mathrm{Ll}$ cascades were found to be $W\left(\theta \right)=1+(4.13\mathrm{}\pm 0.36)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$, $W\left(\theta \right)=1+(1.22\mathrm{}\pm 0.43)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$, and $W\left(\theta \right)=1+(23.32\mathrm{}\pm 2.06)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$, respectively. The results for the $K-L\alpha$ cascade demonstrate the contribution of magnetic quadrupole radiation to the x-ray transitions. When this contribution is taken into account, relatively good agreement with theory is obtained in all cases. Better agreement would be obtained for slightly higher (than theoretically calculated) $M_2$ contributions to the $L$ x-ray transitions.

The atomic absorption cross section of barium has been measured in the autoionization region at an instrumental bandwidth of 0.075 Å between 2380 and 2100 Å and 0.75 Å between 2100 and 1700 Å. The cross sections of the absorption features whose half-widths approach the bandwidth of the monochromator have been corrected to eliminate the bandwidth dependence. The observed structure is grouped into series and presented with measured wavelengths and a parameter $F$. The latter is numerically equal to the oscillator strength of the transition, except when significant perturbations from neighboring lines or the underlying continuum occur. Evidence of the presence of elements in the series $6pns{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{} \mathrm{}(n>\mathrm{}6\mathrm{})\mathrm{}$ is presented.

The propagation of light in an optically pumped vapor is determined semiclassically by the expectation value of a dielectric susceptibility operator. We present explicit formulas for the dielectric susceptibility of a vapor of alkali atoms. The qualitative features of the susceptibility are illustrated for the specific case of ${\mathrm{Rb}}^{87}$. The theory of this paper is essential for a clear understanding of phenomena such as the birefringence of alkali vapors and the microwave modulation of light by alkali vapors. Some experimental observations of birefringence in rubidium vapor are interpreted with the semiclassical theory.

A variational formulation of Brueckner's theory has been applied to the calculation of hyperfine parameters for the $2{}_{}{}^2{}_{}{}^{}P$ excited state of atomic lithium. The hierarchy of $n$th-order (or $n$-particle) Bethe-Goldstone equations defined previously has been modified so that complete electronic configurations occur at each level of the hierarchy. Thus ${\overrightarrow{\mathrm{L}}}^2$, ${\overrightarrow{\mathrm{S}}}^2$ eigenfunctions could be used throughout the calculations, although for practical reasons such functions are not explicitly constructed. Each net increment of any mean-value electronic property defined within the hierarchy is shown to be equivalent to a sum, to infinite order, of a certain well-defined set of linked diagrams in the many-particle perturbation theory. Computed results are in excellent agreement (within roughly 1%) with experimental data and with a previously published accurate perturbation calculation.

Raman scattering of light from representative alkali-halide crystals containing C${\mathrm{N}}^{-}$, N$\mathrm{O}_2^{}{}_{}{}^{-}$, O${\mathrm{H}}^{-}$, and O${\mathrm{D}}^{-}$ impurities is reported and analyzed. The observed spectra have a low-frequency range in which the scattered light is usually shifted from the incident light by less than 300-400 ${\mathrm{cm}}^{-1\mathrm{}}$, and a high-frequency range in which the shifts are typically 1000-2000 ${\mathrm{cm}}^{-1\mathrm{}}$. Although the low-frequency region does not readily lend itself to quantitative analysis, it is clear that its main features can be interpreted in terms of a mixture of second-order scattering from the pure host, impurity-induced first-order scattering that results from perturbing the pure host, and scattering from the rotational degrees of freedom of the molecular impurity. The high-frequency region, on the other hand, consists of spectra whose frequencies are characteristic of the internal normal coordinates of the molecule. A very narrow totally polarized line with depolarized sideband structure is generally observed. The sharp central component is at the frequency of an internal molecular normal coordinate and, typically, has a linewidth of 1 ${\mathrm{cm}}^{-1\mathrm{}}$. It is not significantly affected by the type of host or changes in temperature. It is found that the sideband structure gives a measure of the molecular rotational dynamics. Depending on host and impurity, the observed characteristic behavior varies from nearly free rotation to heavily trapped librational motion. The techniques employed here, both theoretical and experimental, demonstrate and define the usefulness of the Raman effect in studying systems of an analogous nature.